1. **State the problem:** We need to find the unit vector and direction of a given vector $\mathbf{v}$. The direction is the angle the vector makes with the positive x-axis. 2. **Find the magnitude of the vector:** Given $\mathbf{v} = \langle v_x, v_y \rangle$, its magnitude is $$|\mathbf{v}| = \sqrt{v_x^2 + v_y^2}.$$ This is essential for finding the unit vector. 3. **Find the unit vector:** The unit vector in the direction of $\mathbf{v}$ is $$\hat{u} = \frac{\mathbf{v}}{|\mathbf{v}|} = \left\langle \frac{v_x}{|\mathbf{v}|}, \frac{v_y}{|\mathbf{v}|} \right\rangle.$$ It has length 1 and points in the same direction as $\mathbf{v}$. 4. **Find the direction angle $\theta$:** The angle $\theta$ the vector makes with the positive x-axis is given by $$\theta = \tan^{-1}\left(\frac{v_y}{v_x}\right).$$ This angle indicates the direction. **Final answers:** - Unit vector: $$\hat{u} = \left\langle \frac{v_x}{\sqrt{v_x^2 + v_y^2}}, \frac{v_y}{\sqrt{v_x^2 + v_y^2}} \right\rangle.$$ - Direction angle: $$\theta = \tan^{-1}\left(\frac{v_y}{v_x}\right).$$